Subsequences and blocks of Schauder bases

Question

Suppose $X$ is a Banach space and $(e_n)$ and $(f_n)$ are both Schauder bases of $X$.

1. Does there exist a proper closed subspace $Y\subset X$, and appropriate subsequences of $(x_n)$ and $(y_n)$ that are bases for $Y$?
2. Does there exist a proper closed subspace $Y\subset X$, and appropriate block subsequences of $(x_n)$ and $(y_n)$ that are bases for $Y$?